Solution :
Yes, that's a small bug (not exactly a bug though, one can take it as a security mechanism) and it doesn't allow you to send apk via Stock File Manager. However, if you try to send the application package file via other file manager apps like ES File Explorer or Root Explorer or Astro File Manager then, you can easily send those apps. I am sure you can find either of these in the market for free. So, try it out and it will work.

Solution 2 :
Also, another trick is to rename the application package file to .bak extension (or you can append .bak to .apk making it something like application_name.apk.bak temporarily) and then send it to a desired device. One sent ask the receiver to rename it back to .apk and he will be able to use it as normal package. I've not tried it but, it should work fine since the limit is on file extension and in our case we've modified it.